Suppose a_n∈ [0,1] and X_n is a sequence of i.i.d random variables with p.d.f : p(X_n=1)= p(X_n= -1)=0.5 . ∑_(n=1)^∞▒a_n X_n is convergent with probability 1, is ∑_(n=1)^∞▒a_n^2 convergent?

Question

Suppose  a_n∈ [0,1] and X_n is a sequence of i.i.d random variables with p.d.f : p(X_n=1)= p(X_n= -1)=0.5 . ∑_(n=1)^∞▒a_n  X_n is convergent with probability 1, is ∑_(n=1)^∞▒a_n^2   convergent?